부산대학교 도서관

A group $G$ is {\it amenable} if there exists a finitely additive translation invariant probability measure on all subsets of $G$. (This definition from 1929 is due to J. von~Neumann, and there are equivalent combinatorial definitions in terms of the Cayley graph of $G$.) The class of elementary amenable groups is the smallest class of groups that contains all finite groups, all abelian groups, and is closed under taking subgroups, quotients, extensions, and directed unions. Von~Neumann had observed that all elementary amenable groups are amenable, but it was some time before any other examples of amenable groups were found. Chou proved in 1980 that elementary amenable groups have polynomial or exponential growth, and the much studied {\it Grigorchuk group} from 1983, which acts on a rooted tree, is amenable but has intermediate growth, so it cannot be elementary amenable. \par It was proved in [L. Bartholdi, V.~A. Kaimanovich and V.~V. Nekrashevych, Duke Math. J. {\bf 154} (2010), no.~3, 575--598; MR2730578] that groups acting on rooted trees with {\it bounded activity} are amenable; these are the so-called {\it bounded automata groups}. K. Juschenko proved the first results concerning elementary amenable groups that act on rooted trees in [J. Topol. Anal. {\bf 10} (2018), no.~1, 35--45; MR3737508]. \par The question addressed in the paper under review is: ``What do elementary amenable bounded automata groups look like? How close are they to abelian?'' The questions are answered for three large classes of bounded automata groups, which contain many of the examples that have been previously studied. Two of these classes are new. In addition, several examples are presented that place restrictions on what ``close to abelian'' might mean. \par These three classes are iterated monodromy groups, generalized basilica groups, and groups of abelian wreath type (which contain the Gupta-Sidki groups). The principal results proved are: \roster \item"(i)" Every iterated monodromy group of a post-critically finite polynomial is either virtually abelian or not elementary amenable. \item"(ii)" Every generalized basilica group is either (locally finite)-by-(virtually abelian) or not elementary amenable. \item"(iii)" Every self-replicating group of abelian wreath type is either virtually abelian or not elementary amenable. \endroster \par In addition, the concept of an {\it odometer} in a group is introduced, and it is proved that every bounded automata group that contains an odometer is either virtually abelian or not elementary amenable. \par The paper is long but generally well written and reasonably self-contained.